When I cut huge images, AI literally puts them in clipboard and it takes minutes to complete and sometimes crashes AI.

But I usually do this only to immediately paste them in front/back of other objects, to make desired order.
AI don't have commands in Layers palette like 'Move in front/back of selected objects', so this is almost the only way to achieve this.
Dragging selected objects (including images) within Layers palette is very incovinient, because I don't know exactly where targeted object is located there (but I can clearly see it on page).
I also can cut image from one file to paste in in next document. To avoid possible crashing, I have to drag-n-drop it, and it's very slow.

Proposed solution — while staying in AI, cut-copy-paste proxies instead of actual pixels.